My questions are about both CP & LRV formulations in FMMs. (1) Dr. Clark used LRV formulation to explain FMMs in her dissertation, I guess her logic behind that is making use of the equivalence of the two formulations by assuming residual variance=1? (2)If yes,can the same logic be used for FMMs in both ML and Bayes estimation no matter Logit or probit link? (3) Is CP associated with ML and Bayes, and LRV associated with WLS? (4) LRV in Dr.Muthen's 2002 paper is slightly different with LRV in Clark's paper. The threshold in Dr.Muthen's paper appeared in the threshold model (i.e. y=1, when y*>threshold...). The threshold in Clark's paper appeared in the equation y*=threshold+loading*FactorScore + residual (y=1 when y*>0...). Are they equivalent formulation? Is there a reason for the difference?

1-2. The latent response (u*) formulation can be used for both logit and probit, where with probit the residual variance is fixed at 1 and with logit fixed at pi-square/3. Mixtures such as FMM uses either ML with logit (I guess ML can use probit link as well) or Bayes with probit.